4. Definition of Infinite Limit: Let X CR, f: X R and a e X'. If for every M > 0 there exists o > 0 such that |f(r)| > M whenever r EX and 0 < |r - al < 6 then we say that the limit as r approaches a of f(x) is oo which is denoted as lim f(r) = o. Suppose a € R, e > 0, and f, g : N*(a, e) → R. If lim f (x) = L > 0 and lim g(x) = 0, prove lim(fg)(x) = 00 %3D エ→a 全→a エ→a

Q: 2. Suppose X C R, a € X', f,g : X → R, lim f(r) = L, and lim g(r) = M. Use the definition of limit…

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Q: 1. Let f(x,y) = x² + y², g(t) = (3t+1, 2t − 3), and F(t) = (fog)(t). Evaluate F'(t) both directly…

A: 1. Given,              f(x, y)=x2+y2, g(t)=3t+1, 2t-3 and F(t)=f∘gt.

Q: Theorem 5.2.3 Let K be a subset of R, let X be any metric space, and let f: K - X be a function.…

A: Given f:K→X be a function, where K⊂ℝand X be any metric space with metric d. Let c∈ℝ and for some…

Q: 5. Let a, b, and e be real numbers with a <e<b and let f: (a, b) R be a monotone function. If e is a…

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Q: 2. a) Find an example of a real valued function f:R +R and SCR so that S is open while the forward…

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Q: I. Find the length of the following sets: 1. e((-0.1,1.5)) 2. P(R) 3. e((-∞,4]) 4. e((4,7) U…

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Q: (b) Give an example of a function f: R-R that has a unique fixed 3. Some functions are not…

A: We have to  prove the given  statements

Q: Prove or disprove: If > an is a convergent series with positive terms, then nan → 0.

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Q: Let f: R R be differentiable on some point c ER, then show that 1. f(x+h)-f(x-h) f'(c) = lim %3D 2h…

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Q: Let S=( 2,8] n[ 1. inf(S) = -2 and sup(S) = 8 -2,6) , then. %3D 2. inf(S) = 2 and sup(S) = 8. 3.…

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Q: . Use the definition of limit to prove lim = 2r - 3 = 7.. %3D %3D

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Q: only if 3. Let X and Y be metric spaces. Let f : X → Y be a function. Let (a;) be a Cauchy sequence…

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Q: Let f: X → Y and g : Y Z be uniformly continuous on X and Y, Prove go f: X→ Z is uniformly…

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Q: 2. a) Let f be absolutely continuous on [a, b]. Will the function |f| be absolutely continuous on…

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Q: 6. Let X be a metric space. Let f: X → R and g : X - R be real-valued Lipschitz functions with…

A: We have to solve given questions.

Q: 4. Let f be an integrable function on |0, 1|. a) Which of the following inequalities is true: i) f…

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Q: 3n-3 2n+9 n eN equals 3n+1 n+3 a. 0, b. [1,2]. c. {1}. d. {3 }. e. None of these

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Q: Problem 4 (Extending continuous functions). Suppose KC Rm is compact and f: KR" is continuous. Show…

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Q: Let A and B be non-empty sets of real numbers with inf(B) = 2 and sup(A) = ; Then sup(2A – 3B)…

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Q: 1. Straight-line contractions. can be said about a and 67 satisfies the following conditions: /…

A: We have to solve the problems

Q: Use the definition to prove that the function in f(x) = 2x - 3 is uniformly continuous.

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Q: 1 1 1. Let f : R →R with f(x) = ;r+ Show that f is a contraction mapping 3* 6. on R using the…

A: Given- Let f : ℝ → ℝ with fx = 13x + 16.  To Show- Show that f is a contraction mapping on ℝ using…

Q: Suppose a₁ > a2 > 0. For n ≥ 2, set an+1 (a) {a2k+1} is decreasing and {a2k} is increasing. (b) {n}…

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Q: Suppose f,9 : D R are continuous on D. uniformly continuous on D. Prove that f +g is uniformly

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Q: 5. Let g be a nonnegative function which is integrable over a set E. Show that for given e > 0 there…

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Q: Let f: R2R where f(x, y) = x² + y² if both x and y are rational and f(x, y) = 0 otherwise. Prove…

A: Let function f:ℝ2→ℝ defined such that, f(x, y)=x2+y2,x, y∈ℚ0,otherwise To prove that Df0, 0 exists.…

Q: 3. Assume that f is continuously differentiable on the interval [a, b]. Show that f is of bounded…

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Q: Let f be the function Jo if x +1 and f(x) = 42 if x = 1. Prove that f is integrable on [0, 2] and…

A: Given :f is  a function such that               fx=0      if x≠1 and42    if x=1Now, on the interval…

Q: Assume f and g are functions that are both integrable on the closed interval [a, b]. Prove that if…

A: Given f and g are integrable functions on the closed interval [a,b]. And f(x)≤g(x) for all x∈[a,b]…

Q: Let f : R- (0} R be such that the sequence (rnf(rn)) converges to 3 whenever (rn) is a sequence of…

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Q: Calculate lim dt.

A: To evaluate this integral , we will use the fundamental theorem of calculus and Lebnitz theorem.

Q: Give two (or more!) examples of sequences (a„) such that a, → 0 but E1 an diverges. Give two…

A: We have to give two examples of sequences an that an→0 but ∑n=1∞an diverge and we have to give two…

Q: Applying Bernolli's inequality, we have 1. (1+ 2x)" > 1 + 2nx, x > -1 2. (1+ 2x)" > 2 + nx, x > -1 3…

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Q: Definition: A set X C R is said to be disconnected if there exists disjoint open sets U and V such…

A: Given the function, f:ℝ→ℝ and the set X⊆ℝ is connected. To show that f(X) is connected: Suppose that…